Matrix equations of hydrodynamic type as lower-dimensional reductions of Self-dual type $S$-integrable systems

0.34 MB·English

by  A. I. Zenchuk

Matrix equations of hydrodynamic type as
lower-dimensional reductions of Self-dual type
S-integrable systems
A.I. Zenchuk
8
Center of Nonlinear Studies of L.D.Landau Institute for Theoretical Physics
0
0
(International Institute of Nonlinear Science)
2
n Kosygina 2, Moscow, Russia 119334
a
E-mail: [email protected]
J

1
1 February 1, 2008

]
I
S
. Abstract
n
i
l
n We show that matrix Q Q Self-dual type S-integrable Partial Differential Equations
×
[ (PDEs) possess a family of lower-dimensional reductions represented by the matrix Q

×
3 n0Q quasilinear first order PDEs solved in [29] by the method of characteristics. In turn,
v these PDEs admit two types of available particular solutions: (a) explicit solutions and
0
(b) solutions described implicitly by a system of non-differential equations. The later
5
0 solutions, in particular, exhibit the wave profile breaking. Only first type of solutions
2 is available for (1+1)-dimensional nonlinear S-integrable PDEs. (1+1)-dimensional N-
.
8 wave equation, (2+1)- and (3+1)-dimensional Pohlmeyer equations are represented as
0
examples. We also represent a new version of the dressing method which supplies both
7
0 classical solutions and solutions with wave profile breaking to the above S-integrable
: PDEs.
v
i
X
r 1 Introduction
a
There are several types of nonlinear equations of Mathematical Physics, which are referred
to as integrable Partial Differential Equations (PDEs). We underline three following types of
equations:
1. Equations integrable by the Inverse Spectral Transform Method (ISTM), which are also
called soliton or S-integrable equations. Korteweg-de-Vries equation (KdV) has been
discovered as the first representative of this class [1]. Among other methods of study of
soliton equations we underline so-called dressing method, which was invented in [2, 3] and
developed in set of papers, for instance, [4, 5, 6, 7, 8], see also [9, 10, 11].
2. Nonlinear PDEs linearizable by some direct substitution, or C-integrable equations [12,
13, 14, 15, 16, 17]. Mostly remarkable is Hoph substitution allowing to integrate Bu¨rgers
type nonlinear PDEs.
1
3. Quasilinear first order PDEs integrable by the method of characteristics [18] and by
generalized hodograph method [19, 20, 21]. The later method is well applicable to (1+1)-
dimensional nonlinear PDEs. Investigation of the higher dimensional systems by this
method requires special efforts(see, forinstance, themethodof hydrodynamic reductions,
which will be cited below).
Many other method for study of the nonlinear PDEs have been developed. For instance,
the algebraic-geometrical approach is well suited for the construction of local solutions [22, 23].
However, only restricted class of global solutions may be described in this way. A method
for implicit description of solutions manifold is the method of hydrodynamic reductions [24,
25, 26, 27, 28], where solutions of multidimensional integrable PDEs are described in terms of
Riemann invariants of some (1+1)-dimensional first order quasilinear PDEs. The criterion for
applicability of such technique to the given nonlinear PDE has been invented. This method
allows to implement infinitely many arbitrary functions of single argument into solution of the
nonlinear PDE satisfying this criterion.
Our approach to theproblem ofconstruction of theparticular solutions to the Self-dualtype
S-integrable PDEs is most similar to the method of hydrodynamic reductions. However, if it
is applicable, we are able to describe the bigger solutions space. Namely, number of arguments
in the arbitrary functions implemented into solution has no restriction and is defined by the
particular nonlinear PDE. But we have no criterion for applicability of this algorithm to the
given nonlinear PDE, unlike the method of hydrodynamic reductions.
We exhibit a class of solutions with wave profile breaking to the S-integrable PDEs selected
above. This phenomenon is not surprising, since, simultaneously, these solutions are solutions
of the well established class of first order quasilinear PDEs in lower dimension, which are known
to possess solutions spaces implicitly described by systems of non-differential equations [29, 30].
Remark that solutions with breaking of wave profile to dispersion-less Kadomtsev-Pertviashvili
equation (dKP) (which is compatibility condition of appropriate pair of vector fields) have been
obtained and described in [31] using ISTM.
In the rest of Introduction we, first of all, recall the method of characteristics applicable to
a certain type of matrix first order quasilinear PDEs, Sec.1.1. Then we remind some important
properties of theSelf-dual type nonlinear S-integrable PDEs, Sec.1.2. Basic novalties appearing
in this paper are underlined in Sec.1.3. Finally, we represent the brief contents of the paper in
Sec.1.4.
1.1 First order quasilinear PDEs integrable by the method of char-
acteristics
As it was mentioned above, we associate the given Self-dual type S-integrable nonlinear PDE
with the family of lower dimensional first order matrix PDEs integrable by the method of
characteristics. Let us describe this family of PDEs. Throughout this paper we will write
superscripts in parentheses in order to distinguish them from powers.
It is well known that the scalar first order PDE
N
u + u ρ(k)(u) = ρ(u), u : Rn+1 R, (1)
t x
k →
k=1
X
can be solved by the method of characteristics [18]. For instance, solution u(x,t) of (1) for
2
ρ = 0 is defined implicitly by the non-differential equation
u = f x ρ(1)(u)t,..,x ρ(N)(u)t . (2)
1 N
− −
Vector generalizations of equation(cid:0) (1), i.e. the systems of seve(cid:1)ral coupled first order scalar
PDEs, have been investigated in set of papers [19, 21, 32, 33] using generalized hodograph
method in most cases. Recently [29] the matrix generalization of eq.(1) has been solved using
algebraic approach:
N
w + w ρ(k)(w) = ρ(w)+[w,Tρ˜(w)], (3)
t x
k
k=1
X
wherew istheunknownQ QmatrixfunctionoftheN+1independent variables(x ,..,x ,t)
1 N
RN+1, T is any constant d×iagonal matrix, [ , ] is the usual commutator between matrices, an∈d
· ·
ρ(k), ρ, ρ˜ : R R, k = 1,..,N are N + 2 arbitrary scalar functions representable by the
→
positive power series:
ρ(j)(z) = α(j;i)zi, ρ(z) = α(i)zi, ρ˜(z) = α˜(i)zi, (4)
i≥0 i≥0 i≥0
X X X
where α(j;i), α(i) and α˜(i) are scalar constants, so that the quantities ρ(k)(w), k = 1,..,N, ρ(w)
and ρ˜(w) are well-defined functions of the matrix w.
Remark. We may always apply the following change of variable t and field w
N
∂ + α(j;0)∂ ∂ , w e−α˜(0)Ttweα˜(0)Tt (5)
t xj → t →
j=1
X
which is equivalent to putting
α(j;0) = 0, α˜(0) = 0 (6)
in the eqs.(4) without loss of generality.
Solutionsspace to thematrixequation (3) withρ = 0is implicitly described by thefollowing
algebraic equation:
w = (7)
αβ
Q
e−Tαρ˜(w)tF x I ρ(1)(w)t,...,x I ρ(N)(w)t eTγρ˜(w)t ,
αγ 1 N
− − γβ
Xγ=1(cid:16) (cid:16) (cid:17) (cid:17)
α = 1,...,Q, β = 1,...,Q, I is Q Q identity matrix.
×
If ρ = 0, then eq.(3) has infinitely many commuting flows. In this case, it is convenient to
replace eq.(3) by the next equation
N
w + w ρ(mk)(w) = [w,Tmρ˜(m)(w)], m = 1,2,..., (8)
tm xk
k=1
X
where index m enumerates commuting flows, ρ(mk) are arbitrary scalar functions representable
by a positive power series
ρ(mj)(z) = α(mj;i)zi, ρ˜(m)(z) = α˜(m;i)zi, (9)
i≥0 i≥0
X X
3
α(mj;i), α˜(m;i) are scalar constants, T(m) are diagonal matrices. Note, that the transposed
equation
N
w˜ + ρ(mk)(w˜)w˜ = [ρ˜(m)(w˜)Tm,w˜], w˜ = wT, m = 1,2,... (10)
tm xk
k=1
X
is also integrable.
Equation (3) as well as its generalizations whose solutions spaces may be completely de-
scribed implicitly by the system of non-differential equations will be referred to as PDE0 in
this paper for the sake of brevity.
Using either the algebraic manipulations (Sec.2) or the modified version of the dressing
method (Sec.3), we will show that PDE0s (8) may be viewed as lower dimensional reductions
of the classical Self-dual type S-integrable PDEs. These PDEs are referred to as PDE1s in this
paper.
1.2 Family of PDE1s
PDE1s may be viewed as compatibility conditions of the following overdetermined system of
linear PDEs for some spectral function V(λ;x) (spectral system):
N N˜ Nm(ma)x
V (λ;x)+ λnmiV (λ;x)+ λn˜miV(λ;x)T(i) = λiV(λ;x)q(mi)(x), (11)
tm xi
i=1 i=1 i=0
X X X
m = 1,2.
Here all functions are Q Q matrix functions, λ is a complex spectral parameter, n and
mi
n˜ are some integers, N×(m) = max(n ,n˜ , i = 1,...,N, i = 1,...,N˜); q(mi)(x) are
mi max mi1 mi2 1 2
some matrix functions of x = (x ,x ...,t ,t ...). Throughout this paper we use Greek letters in
1 2 1 2
the lists of arguments for spectral parameters. The feature of this system is that its LHS is a
first order differential operator having coefficients which are polynomial in λ and independent
on the vector parameter x, i.e. x-dependent potentials appear only in the RHS, which is
also polynomial in λ. Compatibility condition of any pair of eqs.(11) results in PDE1 for
the functions q(mi). Note that eq.(11) uses only multiplication by the spectral parameter λ and
does not involve derivatives of the spectral function with respect to the spectral parameter. For
this reason such S-integrable PDEs as dispersion-less Kadomtsev-Petviashvili equation (dKP)
and Heavenly equation may not be considered in the framework of the represented dressing
algorithm.
Remark, that the spectral system (11) may be replaced by the equivalent one
N N˜
∂ V(λ;x)+ Anmi(λ,ν) ∂ V(ν;x)+ An˜mi(λ,ν) V(ν;x)T(i) = (12)
tm ∗ xi ∗
i=1 i=1
X X
Nm(ma)x
Ai(λ,ν) V(ν;x)q(mi)(x),
∗
i=0
X
where function A(λ,µ) replaces multiplication by λ and we define operator Ai as follows:
Ai = A A. (13)
∗···∗
i
| {z }
4
Emphasise that A is independent on x. The eq.(12) reduces to the eq.(11) if A(λ,ν) = λδ(λ
−
µ)I, where I is Q Q identity matrix.
×
Example 1: N-wave system. Linear spectral system reads in the simplest case:
∂ V(λ;x) A(λ,ν) V(ν;x)T(m) = V(λ;x)[T(m),w(0)(x)], m = 1,2. (14)
tm − ∗
Compatibility condition of this system yields the next PDE1:
[T(1),w(0)] [T(2),w(0)]+[[T(2),w(0)],[T(1),w(0)]] = 0. (15)
t2 − t1
(0) (0)
Most physical applications require reduction w = w¯ , where α = β and ”bar” means
αβ βα 6
complex conjugate. This equation is well known as (1+1)-dimensional N-wave equation [10].
We consider eq.(15) without reductions.
Example 2: Pohlmeyer equation. Another example is assotiated with the following spec-
tral system:
∂ V(λ;x)+A(λ,ν) ∂ V(ν;x) = V(λ;x)w(0)(x), m = 1,2. (16)
tm ∗ xm xm
Compatibility condition of this system yields the next PDE1:
w(0) w(0) = [w(0),w(0)], (17)
x1t2 − x2t1 x1 x2
which may be written in a different form:
(J−1J ) (J−1J ) = 0, (18)
t1 x2 − t2 x1
w(0) = J−1J , w(0) = J−1J .
x1 t1 x2 t2
This is Pohlmeyer equation [34, 35, 36]. Most applications in Physics is assotiated with its
reduction J+ = J, detJ = 1, which yields (Anty)Selfdual Yang-Mills equation (ASDYM).
±
This equation has been studied by many authors: [37, 38, 39, 40, 41, 42, 43, 44] (instanton
solutions), [45] (merons), [46, 47] (finite gap solutions). (2+1)-dimensional version of ASDYM
assotiated with the reduction
∂ J = ∂ J, ∂ w(0) = ∂ w(0) (19)
x2 t1 x2 t1
is also well known:
w(0) w(0) = [w(0),w(0)] or (20)
t1t1 − t2x1 t1 x1
(J−1J ) (J−1J ) = 0.
t1 t1 − t2 x1
It has been studied in [48, 49] (Initial Value Problem), [50, 51] (localized solutions). Here we
will deal with eqs.(17) and (20a) without any reduction for the field w(0).
5
1.3 Basic novalties
There are two manifolds of particular solutions to PDE1s. First manifold is assotiated with the
uniquely solvable linear integral equation for some spectral function (Zakharov-Shabat method
¯
[52], classical ∂-problem [53]). We concentrate on the second manifold of solutions which is
described implicitly by a system of non-differential equations (see Sec.2). Simultaneously, this
manifold is solution manifold to appropriate lower dimensional PDE0. Proper choice of initial
data leads to wave profile breaking. Such solutions for Pohlmeyer equation will be discussed.
In turn, second manifold of solutions has a sub-manifold of explicit solutions. Regarding (1+1)-
dimensional N-wave equation, only such solutions from the second manifold are available. It is
likely that the same conclusion is valid for any other (1+1)-dimensional S-integrable model.
Itisremarkable, thatthereisaversion ofthedressing method, whichjoinsbothmanifoldsof
solutions to DPE1s, Sec.3. This fact is remarkable, because the second manifold is beyond the
scope ofthe classical dressing method. Inaddition, this fact demonstrates that two significantly
different classes of PDEs (S-integrable PDE1s and integrable by the method of characteristics
PDE0s) arecombined by thedressing method. There areexamples ofsimilar combinations. For
instance, dressing method combining S- and C-integrable models has been suggested in [54];
dressing method combining C-integrable equations and equations integrable by the method of
characteristics has been proposed in [30].
All these examples tell us about flexibility of the dressing method which is important for
development of the uniform method for integration of multidimensional nonlinear PDEs.
Usually, we will use notations PDE1(t ,t ;w(0)), PDE0(t ;n ,w) and PDE0(t ;k ,w˜)
m1 m2 m 0 m 0
with arguments reflecting field as well as derivative(s) of field with respect to variable(s) t
m
appearing in PDE. Of course, PDE0s and PDE1s involve also derivatives with respect to x .
i
However, we do not represent them in the lists of arguments. Here w(x) is n Q n Q matrix
0 0
×
and w˜(x) is k Q k Q matrix block functions:
0 0
×
w(0)(x) w(1)(x) w(n0−2)(x) w(n0−1)(x)
···
I 0 0 r(1)
 ··· 
w(x) = 0 I 0 r(2) , (21)
···
 .. .. .. .. .. 
 . . . . . 
 
 0 0 I r(n0−1) 
 ··· 
 w˜(0)(x) I 0 0 
− ···
w˜(1)(x) 0 I 0
 − ··· 
. . . . .
w˜(x) = .. .. .. .. .. , w˜(0) = w(0),
 
 w˜(k0−2)(x) 0 0 I 
 − ··· 
 w˜(k0−1)(x) r˜(1) r˜(2) r˜(k0−1) 
 − ··· 
 
where I and 0 are Q Q identity and zero matrices, w(i) and w˜(i) are Q Q matrix functions,
× ×
n and k are arbitrary integers, r(i), r˜(i), i = 1,...,n 1 are arbitrary constant Q Q
0 0 0
− ×
matrices (if T(i) = 0 in eqs.(25,26)) or arbitrary diagonal Q Q matrices (if T(i) = 0). Each
× 6
PDE1(t ,t ;w(0)) written for the matrix field w(0) and involving derivatives with respect to t
1 2 1
and t possesses a family of PDE0(t ;n ,w) and PDE0(t ;k ,w˜), m = 1,2, n ,k = 1,2,...,
2 m 0 m 0 0 0
as lower dimensional reductions. Of course, PDE0(t ;n ,w) is compatible with PDE0(t ;n ,w),
1 0 2 0
as well as PDE0(t ;n ,w˜) is compatible with PDE0(t ;n ,w˜).
1 0 2 0
For instance, in Sec.2 we will derive the following PDE0s assotiated with N-wave equation
6
(Sec.2.1.1):
w [w,T(m)w] = 0, (22)
tm − n0
w˜ [w˜T˜(m),w˜] = 0, m = 1,2,
tm − k0
T(m) = diag(T(m) T(m)), T˜(m) = T(m)
n ··· n − n
n
and PDE0s assotiated with Pohlmeyer equatio|n (Se{cz.2.1.2}):
w +w w = 0, (23)
tm xm
w˜ +w˜w˜ = 0, m = 1,2.
tm xm
If reduction (19) is imposed, then eq.(23) reduces to
w +w w = 0, w +w w = 0, (24)
t1 x1 t2 t1
w˜ +w˜w˜ = 0, w˜ +w˜w˜ = 0.
t1 x1 t2 t1
General forms of PDE0(t ;n ,w) and PDE0(t ;k ,w˜), m = 1,2, which are lower dimensional
m 0 m 0
reductions of some PDE1(t ,t ;w(0)) are following:
1 2
N N˜
w + w ρ(mi)(w) = [w,T(i)ρ˜(mi)(w)], m = 1,2, (25)
tm xi n0
i=1 i=1
X X
N N˜
w˜ + ρ(mi)(w˜)w˜ = [ρ˜(mi)(w˜)T˜(i),w˜], m = 1,2, (26)
tm xi k0
i=1 i=1
X X
where functions ρ˜(mi) are defined by the series similar to the eq.(9):
ρ˜(mj)(z) = α˜(mj;i)zi, (27)
i≥0
X
and α˜(mj;i) are scalar constants. Remarks on the origin of eqs.(25) and (26) will be given in
Sec.3.3, see n.3,5 and n.4,6 respectively. Eq.(26) has the form of transposed equation (25) with
replacements
w˜ = wT, T(m) T˜(m). (28)
n0 → k0
Because of this similarity, we will deal with PDE1(t ,t ;w(0)) and PDE0(t ;n ,w), m = 1,2.
1 2 m 0
Regarding PDE0(t ;k ,w˜), only some details will be given.
m 0
Remark 1. Although any PDE1 is compatibility condition of eqs.(12), we are not able to
represent general form of PDE1s explicitly, unlike general form of PDE0, eqs.(25,26).
Remark 2. The eq.(25) defers from the eq.(8) by the sum in the RHS. However, eq.(25) may
be treated by the methods developed in [29, 30] for eq.(8).
Remark 3. The eqs.(22,23,25,26) (where m = 1,2) have infinitely many commuting flows
corresponding to m > 2 in these equations.
7
1.4 Brief contents
In the next section (Sec.2) we give algebraic description of PDE1(t ,t ;w(0)), PDE0(t ;n ,w)
1 2 m 0
and PDE0(t ;k ,w˜). Derivation of PDE1 (namely, (1+1)-dimensional N-wave and (3+1)- and
m 0
(2+1)- dimensional Pohlmeyer equations) assotiated with appropriate PDE0(t ;n ,w) and
m 0
PDE0(t ;k ,w˜), m = 1,2 is given in Sec.2.1. Assotiated linear overdetermined systems have
m 0
been described therein as well. Using slightly modified results of [29, 30] we represent the non-
differential matrix equations implicitly describing the solutions spaces to the above PDE0s and
assotiated solutions manifolds to the appropriate PDE1s, Sec.2.2. We derive some manifolds of
explicit solutions to N-wave and Pohlmeyer equations and describe solutions with wave profile
breaking for (2+1)- and (3+1)-dimensional Pohlmeyer equations.
A new version of the dressing method describing PDE0s and their relations with PDE1s is
proposedinSec.3. WehavederivedexamplesofPDE0sandexamplesofPDE1s(followingSec.2,
N-wave and Pohlmeyer equations are taken as examples of PDE1s) together with appropriate
linear overdetermined systems (spectral systems), Sec.3.1. Classical solutions manifolds to
PDE1s (Sec.3.2.1) and solutions manifolds with wave profile breaking to PDE1s and PDE0s
(Sec.3.2.2) have been derived in Sec.3.2.
Although this version of the dressing method is mostly straightforward in spirit of the
derivation of PDE1s, it does not supply full solutions spaces to PDE0s, unlike the method of
characteristics. The second version of the dressing algorithm exhibits fullness of the solutions
spaces to PDE0s, Sec.3.3.
Conclusions are given in Sec.4.
2 Algebraic description of PDE1s and PDE0s
We consider three examples of PDE1(t ,t ;w(0)) (namely, eqs.(15) and (17) together with its
1 2
(2+1)-dimensionalreduction(20a))andappropriatefamiliesofPDE0(t ;n ,w)andPDE0(t ;k ,w˜),
m 0 m 0
m = 1,2, in the framework of the algebraic approach, see Secs.2.1.1,2.1.2. The solutions spaces
to PDE1(t ,t ;w(0)) and PDE0(t ;n ,w), m = 1,2, will be investigated in Sec.2.2.
1 2 m 0
2.1 Derivation of PDE1s and appropriate families of PDE0s
2.1.1 (1+1)-dimensional N-wave equation
It iswell known, that any S-integrableequation can berepresented assystem of two commuting
discrete chains. Mostly explicitly this representation is given in Sato approach to integrability,
see for instance, [55]. However, any known dressing method gives rise to this representation.
We show, that N-wave equation can be derived from the following pair of commuting discrete
chains with two discrete parameters:
w(kn) = w(k(n+1))T(m) T(m)w((k+1)n) +w(k0)T(m)w(0n), m = 1,2, (29)
tm −
supplemented by the next non-differential relation among w(ij):
w((k+1)n) = w(k(n+1)) +w(k0)w(0n) (30)
(see also Sec.3.1.3, eqs.(150,151) with s(m) = 0). In fact, first of all remark, that eq.(29) gives
rise to two alternative discrete chains with single discrete parameter in view of (30). The first
8
chain follows after putting k = 0 and eliminating w(1n) using eq.(30):
w(n) [w(n+1),T(m)]+[T(m),w(0)]w(n) = 0, m = 1,2, (31)
tm −
where
w(n) = w(0n), n = 0,1,.... (32)
The second discrete chain follows after putting n = 0 and eliminating w(k1) from (29) using
(30):
w˜(k) [w˜(k+1),T(m)]+w˜(k)[w˜(0),T(m)] = 0, m = 1,2, (33)
tm −
where
w˜(k) = w(k0), k = 0,1,..., w˜(0) = w(0). (34)
Finally, in order to write PDE1 for the function w(0), we fix n = 0 in (31) and eliminate w(1),
or fix k = 0 in (33) and eliminate w˜(1). In both cases we will end up with the same equation
(15).
Along with PDE1(t ,t ;w(0)) (15) we may derive the family of PDE0(t ;n ,w), m = 1,2,
1 2 m 0
from the eqs.(31) imposing the reduction
n0−1
w(n0)(x) = w(i)(x)r(i), (35)
i=0
X
where n is an arbitrary integer and r(i) are arbitrary diagonal constant matrices. Similarly, we
0
may derive PDE0(t ;k ,w˜), m = 1,2, from the eq.(33) with the reduction
m 0
k0−1
w˜(k0)(x) = r˜(i)w˜(i)(x), w˜(0) = w(0), (36)
i=0
X
where k is arbitrary integer and r˜(i) are arbitrary diagonal constant matrices. Both (35)
0
and (36) are closures of the chains (31) and (33) respectively. However, parameters r(0) and
r˜(0) may be removed from the PDE0s by the shifts of fields w(n0−1) + r(0) w(n0−1) and
→
w˜(k0−1) +r˜(0) w˜(k0−1). Thus hereafter we take
− → −
r(0) = r˜(0) = 0 (37)
without loss of generality. Parameters n , k , r(i) and r˜(i) appear in the definitions of w and w˜,
0 0
eqs.(21). The simplest explicit examples of PDE0(t ;n ,w) and PDE0(t ;k ,w˜) are following:
m 0 m 0
PDE0(t ;1,w) : w(0) +[T(m),w(0)]w(0) = 0, m = 1,2, (38)
m tm
PDE0(t ;2,w), r(1) = 0 : w(0) [w(1),T(m)]+[T(m),w(0)]w(0) = 0, (39)
m tm −
w(1) +[T(m),w(0)]w(1) = 0, m = 1,2,
tm
PDE0(t ;1,w˜) : w˜(0) +w˜(0)[w˜(0),T(m)] = 0, (40)
m tm
PDE0(t ;2,w˜), r˜(1) = 0 : w˜(0) [w˜(1),T(m)]+w˜(0)[w˜(0),T(m)] = 0,
m tm −
w˜(1) +w˜(1)[w˜(0),T(m)] = 0, m = 1,2.
tm
9
It is not difficult to observe that the above PDE0(t ;n ,w) and PDE0(t ;k ,w˜) with arbitrary
m 0 m 0
n and k , may be written in the form (22a) and (22b) respectively, where w and w˜ are defined
0 0
by the formulae (21).
Since the eq. (22b) follows from the eq.(22a) after transposition with replacements (28), it
is enough to study one of them, say, eq.(22a). Slightly modifying result of [30], we describe the
solutions space to the eq.(22a) with arbitrary integer n by the next implicit algebraic equation:
0
wαβ = n0Q e−m2=1(Tn(m0))αwtmFαγ(w) em2=1(Tn(m0))γwtm , (41)
P P γβ
Xγ=1(cid:16) (cid:17)
α,β = 1,...,n Q,
0
where F(z ) is n Q n Q matrix function of single argument. Details of derivation of this
0 0 0
×
formula by the dressing method are given in Secs.3.2.2 and 3.3. However, the structure of w
tells us that F must have the following structure:
arbitrary scalar function of arguments, α Q
F (z ) = ≤ , (42)
αβ 0 δ z , α > Q
αβ 0
(cid:26)
sothatequation(41)becomesanidentityforα > Q. Thus, thesquarematrixalgebraicequation
(41) reduces to the rectangular matrix equation with α = 1,...,Q and β = 1,...,n Q. The
0
scalar functions of single argument F (z ), α = 1,...,Q, β = 1,...,n Q are arbitrary. The
αβ 0 0
simplest case n = 1 yields w w(0), T(m) T(m):
0 ≡ 1 ≡
Q 2 2
wα(0β) = e−mP=1Tα(m)w(0)tmFαγ(w(0)) emP=1Tγ(m)w(0)tm γβ, α,β = 1,...,Q. (43)
Xγ=1(cid:16) (cid:17)
Assotiated linear overdetermined system of PDEs. As it was mentioned above, the
linear spectral problem for N-wave equation is given by the eqs.(14), where V(λ;x) is, generally
speaking, a rectangular lQ Q matrix function, l is some integer (to anticipate, l is assotiated
×
with the dimension of the kernel of the integral operator in eq.(114); l = 2 in Secs.3.1,3.2). One
can extend this spectral system introducing discrete parameter n by the following formulae:
A(λ,ν) V(n)(ν;x) = V(n+1)(λ;x)+V(0)(λ;x)w(n)(x), (44)
∗
V(n)(λ;x) A(λ,ν) V(n)(ν;x)T(m) = V(0)(λ;x)[T(m),w(n)(x)], (45)
tm − ∗
n = 0,1,2,...,m = 1,2.
which becomes eq.(14) if n = 0 and V V(0). This extension can be derived formally, for
≡
instance, using a version of the dressing method developed in Sec.3, see Sec.3.1.3 eq.(149) with
s(m) = 0.
Let us consider the reduction (35,37) for the fields w(n), which causes the appropriate re-
duction for the spectral functions V(n):
n0−1
V(n0) = V(i)r(i). (46)
i=1
X
In view of this reduction, we may introduce the block-vector spectral function
V = [V(0) V(n0−1)]. (47)
···
10